**73**

votes

**4**answers

9k views

### How small can a sum of a few roots of unity be?

Let $n$ be a large natural number, and let $z_1, \ldots, z_{10}$ be (say) ten $n^{th}$ roots of unity: $z_1^n = \ldots = z_{10}^n = 1$. Suppose that the sum $S = z_1+\ldots+z_{10}$ is non-zero. How ...

**69**

votes

**10**answers

9k views

### “Understanding” $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$

I have heard people say that a major goal of number theory is to understand the absolute Galois group of the rational numbers $G = \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$. What do people mean when ...

**46**

votes

**13**answers

5k views

### Erratum for Cassels-Froehlich

Edit 25 April 2010: I have a physical copy of the new printing of the book. I can only assume the LMS is now selling it (but have no details).
IMPORTANT EDIT: THE RESULTS ARE IN! Ok, the deadline has ...

**44**

votes

**5**answers

2k views

### If a unitsquare is partitioned into 101 triangles, is the area of one at least 1%?

Update: The answer to the title question is not necessarily, as pointed out by Tapio and Willie. I would be more interested in lower bounds.
Monsky's famous and amazingly tricky proof says that if we ...

**39**

votes

**7**answers

4k views

### How to picture $\mathbb{C}_p$?

I hope this is appropriate for mathoverflow. Understanding $\mathbb{C}_p$ has always been something of a stumbling block for me. A standard thing to do in number theory is to take the completion ...

**39**

votes

**3**answers

3k views

### What was the relative importance of FLT vs. higher reciprocity laws in Kummer's invention of algebraic number theory?

This question is inspired in part by this answer of Bill Dubuque, in which he remarks that the fairly common belief that Kummer was motivated by FLT to develop his theory of cyclotomic number fields ...

**38**

votes

**2**answers

3k views

### Formal group laws and L-series

Let E be an elliptic curve, let $L(s) = \sum a_n n^{-s}$
denote its L-function, and set
$$ f(x) = \sum a_n \frac{x^n}{n}. $$
Then Honda has observed that
$$ F(X,Y) = f^{-1}(f(X) + f(Y)) $$
defines ...

**37**

votes

**4**answers

3k views

### Fermat's last theorem over larger fields

Fermat's last theorem implies that the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}$ is finite.
Is the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}^{\text{ab}}$ finite?
Here ...

**34**

votes

**13**answers

3k views

### Applications of the Cayley-Hamilton theorem

The Cayley-Hamilton theorem is usually presented in standard undergraduate courses in linear algebra as an important result. Recall that it says that any square matrix is a "root" of its own ...

**34**

votes

**1**answer

1k views

### Degree 17 number fields ramified only at 2

The number $17$ is the smallest odd number that occurs as the degree of a number field $K/\mathbb{Q}$ for which the only finite prime that ramifies is $2$. The non-existence for $n < 17$ follows ...

**33**

votes

**3**answers

2k views

### If Spec Z is like a Riemann surface, what's the analogue of integration along a contour?

Rings of functions on a nonsingular algebraic curve (which, over $\mathbb{C}$, are holomorphic functions on a compact Riemann surface) and rings of integers in number fields are both examples of ...

**33**

votes

**1**answer

1k views

### Is my field algebraically closed?

For a field $L$, let $\widetilde L$ be the splitting field of all irreducible polynomials over $L$ having prime-power degree.
Question: Do we have $\widetilde{\mathbf Q}=\overline{\mathbf Q}$?
...

**33**

votes

**2**answers

1k views

### What is an infinite prime in algebraic topology?

The links between algebraic topology (stable homotopy theory in particular) and number theory are nowadays abundant and fruitful. In one direction, there is chromatic homotopy theory, exploiting the ...

**32**

votes

**3**answers

978 views

### Simple argument regarding sums of two units in a number field?

I wonder if it is possible to show, without using the Schmidt subspace/Roth theorem/Baker's bounds on linear forms in logarithms or other very deep results, that, in a number field, not all integral ...

**31**

votes

**1**answer

3k views

### Integers not represented by $ 2 x^2 + x y + 3 y^2 + z^3 - z $

EDIT, 9 March 2014: when I asked this in 2010, I did not have the courage of my convictions, and so did not ask for an if and only if proof, as Kevin Buzzard quite properly pointed out. Such problems ...

**29**

votes

**3**answers

2k views

### On what kind of objects do the Galois groups act?

I am neither number theorist nor algebraic geometer. I am wondering
whether Galois groups of number fields (say the absolute Galois
group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$) act on objects which
...

**28**

votes

**6**answers

1k views

### Patterns among integer-distance points

Mark each point of $\mathbb{N}^2$ ($\mathbb{N}$ the natural numbers) if its
Euclidean distance from the origin is an integer. One obtains a plot like this, symmetric about the $45^\circ$ diagonal.
...

**28**

votes

**1**answer

2k views

### How would Hilbert and Weber think about the Langlands programme?

Explanations to a general mathematical audience about the Langlands programme often advertise it as "non-abelian class field theory". They usually begin as follows: a modern style formulation of ...

**28**

votes

**2**answers

3k views

### Class Numbers and 163

This is a bit fluffier of a question than I usually aim for, so apologies in advance if this doesn't pass the smell test for suitability.
Likely my favorite fun fact in all of number theory is the ...

**27**

votes

**1**answer

1k views

### What happened to Emmy Noether's *Zukunftsphantasie* ?

Recenly I came across Peter Roquette's article On the history of Artin's $L$-functions and conductors (23 July 2003) in which he talks about some letters from Emil Artin and Emmy Noether to Helmut ...

**26**

votes

**3**answers

2k views

### Why aren't there more classifying spaces in number theory?

Much of modern algebraic number theory can be phrased in the framework of group cohomology. (Okay, this is a bit of a stretch -- much of the part of algebraic number theory that I'm interested ...

**26**

votes

**1**answer

1k views

### Do the algebraic integers form a free abelian group?

It is a well-known fact, proved in every introductory textbook on algebraic number theory, that if $K$ is an algebraic number field, i.e. a finite extension of $\mathbb{Q}$, then its ring ...

**25**

votes

**4**answers

2k views

### $A_5$-extension of number fields unramified everywhere

So I was having tea with a colleague immensely more talented than myself and we were discussing his teaching algebraic number theory. He told me that he had given a few examples of abelian and ...

**25**

votes

**3**answers

2k views

### Interactions between (set theory, model theory) and (algebraic geometry, algebraic number theory ,…)

Set theory and model theory have many applications outside of logic, in particular in algebra, topology, analysis, ...
On the other hand model theory, in particular after Hrushovski, found many ...

**25**

votes

**1**answer

1k views

### Is pi = log_a(b) for some integers a, b > 1?

Are there integers $a, b > 1$ such that $\pi = \log_a(b)$?
Or equivalently: are there integers $a,b > 1$ such that $a^\pi = b$?
Note that the transcendence of $\pi$ makes this a problem - ...

**25**

votes

**3**answers

2k views

### Galois Groups of a family of polynomials

I've stumbled across the family of polynomials
$ f_p(x) = x^{p-1} + 2 x^{p-2} + \cdots + (p-1) x + p $,
where $p$ is an odd prime. It's not too hard to show that $f_p(x)$ is irreducible over ...

**25**

votes

**1**answer

2k views

### Ramanujan and algebraic number theory

One out of the almost endless supply of identities discovered by Ramanujan
is the following:
$$ \sqrt[3]{\sqrt[3]{2}-1} = \sqrt[3]{\frac19} - \sqrt[3]{\frac29} + \sqrt[3]{\frac49}, $$
which has the ...

**25**

votes

**0**answers

914 views

### Derivative of Class number of real quadratic fields

Let $\Delta$ be a fundamental quadratic discriminant, set $N = |\Delta|$,
and define the Fekete polynomials
$$ F_N(X) = \sum_{a=1}^N \Big(\frac{\Delta}a\Big) X^a. $$
Define
$$ f_N(X) = ...

**24**

votes

**10**answers

3k views

### What are some interesting problems in the intersection of Algebraic Number Theory and Algebraic Topology?

I'm a beginning graduate student and while my background is primarily in algebraic number theory, I've found myself a bit smitten with the subject of algebraic topology recently after only having read ...

**24**

votes

**3**answers

811 views

### Intuition for Zagier's theorem for $\zeta_K(2)$

In 1986, Don Zagier generalized Euler's theorem ($\zeta_\mathbb{Q}(2)=\pi ^2 /6$) to an arbitrary number field $K$:
$$\zeta_K(2)=\frac{\pi^{2r+2s}}{\sqrt{|D|}}\times \sum_v c_v ...

**24**

votes

**2**answers

2k views

### How often are irrational numbers well-approximated by rationals?

Suppose $x\in \mathbb{R}$ is irrational, with irrationality measure $\mu=\mu(x)$; this means that the inequality $|x-\frac{p}{q}|< q^{-\lambda}$ has infinitely many solutions in integers $p,q$ if ...

**24**

votes

**0**answers

703 views

### On certain representations of algebraic numbers in terms of trigonometric functions

Let's say that a real number has a simple trigonometric representation, if it can be represented as a product of zero or more rational powers of positive integers and zero or more (positive or ...

**23**

votes

**1**answer

494 views

### Artin reciprocity $\implies $ Cubic reciprocity

I asked this on math.SE a few days ago with no reply, so I'm reposting it here. Hope this is not considered too elementary for MO (feel free to close if so).
I'm trying to understand the proof of ...

**23**

votes

**0**answers

1k views

### Orders in number fields

Let $K$ be a degree $n$ extension of ${\mathbb Q}$ with ring of integers $R$. An order in $K$ is a subring with identity of $R$ which is a ${\mathbb Z}$-module of rank $n$.
Question: Let $p$ be an ...

**22**

votes

**2**answers

787 views

### Elementary congruences and L-functions

In a recent article, Emmanuel Lecouturier proves a generalization of the following surprising result: for a Mersenne prime $N = 2^p - 1 \ge 31$, the element
$$ S = \prod_{k=1}^{\frac{N-1}2} k^k $$
...

**21**

votes

**1**answer

2k views

### Fermat's Last Theorem in the cyclotomic integers.

Kummer proved that there are no non-trivial solutions to the Fermat equation FLT(n): $x^n + y^n = z^n$ with $n > 2$ natural and $x,y,z$ elements of a regular cyclotomic ring of integers $K$.
I am ...

**21**

votes

**1**answer

1k views

### Weil Conjectures for Number Fields

Let $K$ be a number field with integral basis $\{\omega_1,\ldots,\omega_n\}$.
The affine variety $A_K$ defined by
$$ N_{K/{\mathbb Q}}(X_1 \omega_1 + \ldots + X_n \omega_n) = 1 $$
is an algebraic ...

**21**

votes

**0**answers

231 views

### Does every ring of integers sit inside a monogenic ring of integers?

Given a number field $K/\mathbf{Q}$ whose ring of integers $\mathcal{O}_K$ is, in general, not of the form $\mathbf{Z}[\alpha]$ (not monogenic), does there exist an extension $L/K$ which has ...

**20**

votes

**4**answers

3k views

### The Riemann Hypothesis and the Langlands program

On page 263 of this book review appears the following:
Given the centrality of L-functions to the Langlands program, nothing would seem more natural (than a presentation of elementary algebraic ...

**20**

votes

**1**answer

1k views

### What's special about the circle problem?

Let $K$ be a number field, and let
$$\zeta_{K}(s):= \sum_{0
\neq I \text{ ideal of }O_K} \frac{1}{N_{K/\mathbb{Q}}(I)^s} = \sum_{n \ge 1} \frac{a_n}{n^s}$$
be the Dedekind zeta function of $K$. The ...

**20**

votes

**1**answer

485 views

### Proportion of irreducible polynomials $P$ such that $\mathbf Z[X]/(P)$ is the ring of integers of $\mathbf Q[X]/(P)$

I know that number fields have been the object of many statistical experiments.
Is there some kind of heuristics for the following?
Fix a degree $d$ and fix a bound $N$ on the coefficients of a monic ...

**20**

votes

**2**answers

1k views

### Are there any Hecke operators acting on an elliptic curve with additive reduction that I don't know about?

I could have made this question very brief but instead I've maximally gone the other way and explained a huge amount of background. I don't know whether I put off readers or attract them this way. The ...

**19**

votes

**2**answers

880 views

### Motivating Lubin-Tate theory

The Lubin-Tate theory gives an amazingly clean and streamlined way of constructing the subfield (usually denoted) $F_\pi\subset F^\mathrm{ab}$ for a local field $F$ fixed by the Artin map associated ...

**19**

votes

**1**answer

558 views

### Décomposition des nombres premiers dans des extensions non abéliennes

Gauß famously determined the cubic character of $2$ in his Disquisitiones : $2$ is a cube modulo a prime number $p\equiv1\mod3$ if and only if $p=x^2+27y^2$ for some $x,y\in\mathbf{Z}$. This implies ...

**19**

votes

**1**answer

959 views

### What is the ring of integers of the Pythagorean field?

Following Hilbert, we call the complex numbers constructible via
compass and straight-edge the field of Euclidean numbers, and
the totally real such numbers the field of Pythagorean numbers. (Among ...

**19**

votes

**0**answers

830 views

### Most “natural” proof of the existence of Hilbert class fields

Assume that you have proved the two inequalities of class field theory, and that you want to show that the Hilbert class field, i.e., the maximal unramified abelian extension, of a number field $K$ ...

**18**

votes

**2**answers

1k views

### A question about non-norm-euclidean real quadratic fields

After lurking MO for a while, I decided I'd jump in. Why not start with a question?
Specifically, I have a question about a natural set that sort of "measures" the failure of a real quadratic field ...

**18**

votes

**1**answer

709 views

### Concrete Applications of knowing $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$

I have very little experience with Galois representations, mostly as they relate to class field theory, elliptic curves, and modular forms, but they seem to have quite a reputation in number theory as ...

**18**

votes

**3**answers

2k views

### Commutative Algebra with a View Toward Algebraic Number Theory

Someone asked me this today, and I don't know what the standard answer is:
Is there an analogue of David Eisenbud's rather amazing Commutative Algebra With a View Toward Algebraic Geometry but with a ...

**18**

votes

**1**answer

1k views

### Can Eisenstein's lattice point proof of quadratic reciprocity be generalized?

I'm referring to this proof. The key formula ("Eisenstein's Lemma") is
$$\left(\frac{q}{p}\right)=(-1)^{\sum_{u}\lfloor\frac{qu}{p}\rfloor},\text{ where $u=2,4,\ldots,p-1$}$$
The sum in the exponent ...